If `A = tan 11^(@) tan 29^(@), B = 2cot 61^(@) cot 79^(@)`, then which among the following is true?

Options

A. A = 2B
B. A =-2B
C. 2A = B
D. 2A =- B

Answer

To solve the problem, we need to evaluate the expressions for \( A \) and \( B \) and then find the relationship between them.

Given:
\[ A = \tan(11^\circ) \tan(29^\circ) \]
\[ B = 2 \cot(61^\circ) \cot(79^\circ) \]

### Step 1: Simplify \( B \)

Using the complementary angle identity for cotangent:
\[ \cot(90^\circ - \theta) = \tan(\theta) \]

We can rewrite \( \cot(61^\circ) \) and \( \cot(79^\circ) \):
- \( \cot(61^\circ) = \tan(29^\circ) \) (since \( 61^\circ + 29^\circ = 90^\circ \))
- \( \cot(79^\circ) = \tan(11^\circ) \) (since \( 79^\circ + 11^\circ = 90^\circ \))

Now substitute these back into \( B \):
\[ B = 2 \cot(61^\circ) \cot(79^\circ) = 2 \tan(29^\circ) \tan(11^\circ) \]

### Step 2: Relate \( A \) and \( B \)

Now we can express \( A \) in terms of \( B \):
\[ A = \tan(11^\circ) \tan(29^\circ) \]
\[ B = 2 \tan(11^\circ) \tan(29^\circ) \]

### Step 3: Find the relationship

From the above, we can see that:
\[ B = 2A \]

This means:
\[ A = \frac{1}{2} B \]

### Conclusion

Thus, the correct relationship between \( A \) and \( B \) is:
\[ 2A = B \]

### Final Answer

The correct option is:
**Option C: \( 2A = B \)**

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